Linear programs Optimization Geoff Gordon Ryan Tibshirani

Transcription

1 Linear programs Optimization Geoff Gordon Ryan Tibshirani

2 Review: LPs LPs: m constraints, n vars A: R m n b: R m c: R n x: R n ineq form [min or max] c T x s.t. Ax b m n std form [min or max] c T x s.t. Ax = b x 0 m n max 2x+3y s.t. x + y 4 2x + 5y 12 x + 2y 5 x, y 0 2

3 Review: LPs Polyhedral feasible set infeasible (unhappy ball) unbounded (where s my ball?) Optimum at a vertex (= a 0-face) Transforming LPs changing to to = getting rid of free vars or bounded vars 3

4 Review: LPs Tableau: x y u v w z RHS Row operations to get equivalent tableaux Basis (more or less corresponds to a corner) use row ops to make m m block of tableau = identity matrix set nonbasic vars = 0: enough constraints to fully specify all other variables (so, a 0-face, if it s feasible) 4

5 Ineq form is projected std form z x, y, z 0 A[x;y;z] = b y x 5

6 Three bases 1 0 5/7 10/ /7 5/7 x 5 z /7 y 1 1/ / x 5 x 10/7 z y 6

7 What if we can t pick basis? E.g., suppose A doesn t have full row rank can t pick m linearly independent cols Ex: 3x + 2y + 1z = 3 6x + 4y + 2z = 6 7

8 What if we can t pick basis? E.g., suppose fewer vars than constraints A taller than it is wide, m n can t pick enough cols of A to make a square matrix Ex: 8

9 We can assume Nonsingular n m (at least as many vars as constrs) A has full row rank Else, drop rows (maintaining rank) until it s true Called nonsingular standard form LP 9

10 Naive (sloooow) algorithm Put in nonsingular standard form Iterate through all subsets of n vars if m constraints, how many subsets? Check each for full rank ( basis-ness ) feasibility (RHS 0) If pass both tests, compute objective Maintain running winner, return at end 10

11 Improving our search Naive: enumerate all possible bases Smarter: maybe neighbors of good bases are also good? Simplex algorithm: repeatedly move to a neighboring basis to improve objective continue to assume nonsingular standard form LP 11

12 Neighboring bases Two bases are neighbors if they share (m 1) variables Neighboring feasible bases correspond to vertices connected by an edge x y z u v w RHS def n: pivot, enter, exit 12

13 Example max z = 2x + 3y s.t. x + y 4 2x + 5y 12 x + 2y 5 x 4 x y s t u v z RHS

14 x y s t u v z RHS

15 x y s t u v z RHS

16 x y s t u v z RHS

17 Initial basis x y u v w RHS So far, assumed we started w/ feasible basic solution in fact, it was trivial to find one Not always so easy in general 17

18 Big M 0 x, y, s1..s6 max x - 2y x y slacks z RHS Can make it easy: variant of slack trick For each violated constraint, add var w/ coeff 1 Penalize in objective; negate constraint 18

19 Simplex in one slide (skipping degeneracy handling) Given a nonsingular standard-form max LP Start from a feasible basis and its tableau big-m if needed Pick non-basic variable w/ coeff in objective 0 Pivot it into basis, getting neighboring basis select exiting variable to keep feasibility Repeat until all non-basic variables have objective 0 19

20 Degeneracy Not every set of m variables yields a corner some have rank < m (not a basis) some are infeasible Can the reverse be true? Can two bases yield the same corner? 20

21 Degeneracy x y u v w RHS / / / / /

22 Degeneracy in 3D 22

23 Bases & degeneracy How many bases for vertex A? Are they all neighbors of one another? Are they all neighbors of B? A B 23

24 Dual degeneracy More than m entries in objective row = 0 so, a nonbasic variable has reduced cost = 0 objective orthogonal to a d-face for d 1 24

25 Handling degeneracy Sometimes have to make pivots that don t improve objective stay at same corner (exiting variable was already 0) move to another corner w/ same objective (coeff of entering variable in objective was 0) Problem of cycling need an anti-cycling rule (there are many ) e.g.: add tiny random numbers to obj, RHS 25